Since my previous post, I have received a number of comments from people that suggest they don’t understand what mathematics is. Although there is no single, accepted definition, I think it’s important for maths teachers and educationalists to have a fairly clear idea of what maths is otherwise we risk innovating our way out of teaching it.
In short, my claim is that mathematics uses deductive reasoning whereas areas such as science and history use inductive reasoning. This means that mathematical results are certain in a way that scientific or historical truths are not. Instead, the latter are probabilistic. Here is a useful primer on inductive versus deductive reasoning. Critically, the certainty that mathematics provides has a pretty big flaw – truths are only true if the axioms they are based upon are true and mathematics cannot be used to prove its own starting axioms.
To give an example of deductive reasoning:
God favours the English
This woman is English
Therefore, God favours this woman
The logic of the syllogism is complete and flawless, but this does not mean that God favours the English. Such a question is obviously outside the realms of deductive reasoning (note that I am not claiming that all deductive reasoning is mathematics – I am using these syllogisms because they often make the point more clearly than a mathematical example).
This is similar to the supposedly subjective bits of maths that people point me towards, such as an assumption about a base-rate probability or whether a data point is an outlier. These questions are scientific, in the broadest sense. For instance, computing p-values for psychology experiments has become controversial in recent years. Nevertheless, the deductive logic involved in computing them remains the same. The controversy lies in the inferences we draw on the basis of these values and that, again, is a scientific question. We cannot mathematically prove the validity or otherwise of the p-value.
To give another example, suppose we have a sample of 200 people and we know that 5 have influenza, we can then do plenty of maths. We can determine the standard deviation, work out confidence intervals and so on. The axioms we use come from the data and from assumptions that sit behind standard deviations and confidence intervals. We cannot mathematically prove the value of confidence intervals and we cannot mathematically prove any scientific conclusions. However, these values may be used to inform scientific, inductive reasoning.
Essentially, applied maths makes use of maths in answering other kinds of questions. There is a maths part and a non-maths part to this process. If you are a trained scientist in the heat of problem solving, you may not see or even care about the distinction, but there is one.
Clearly, there are plenty of questions that cannot be addressed by mathematics and there are paradoxes within mathematics where mathematical reasoning breaks down (an example of a paradox is that you cannot prove the truth or otherwise of the statement ‘this statement is false’ using deductive reasoning). However, neither of these introduce matters of opinion as part of mathematical reasoning and neither of these falsify my claim that mathematics is right or wrong. That’s like suggesting the claim that ‘a car is a form of transport’ is false because cars sometimes break down or cars cannot transport minke whales.
Similarly, Gödel’s Theorem does not prove the subjectivity of mathematics, despite what some have claimed. Ironically for those who attempt to make this argument, Gödel’s theorem is itself a beautiful piece of deductive reasoning. It demonstrates that whatever formal system of mathematics we devise, there will be mathematical truths that cannot be demonstrated to be true within that system. This provides a theoretical limit on what we can prove within any given formal system. It does not introduce subjectivity.